Algebraic values of complex functions

Transcription

1 Algebraic values of complex functions Marco Streng DIAMANT symposium Heeze May 2011

2 algebraic numbers Definition: α C is an algebraic number if there is a non-zero polynomial f = a n x n + + a 1 x + a 0 Q[x] with f(α) = 0. Examples: 2 is a root of x 2 2, i is a root of x 2 + 1, ± 5, 5 9, etc., etc. all five complex zeros of x x + 5 are algebraic Let Q C be the set of all algebraic numbers. C is uncountable, but Q is only countable e = n=0 1 n! is not algebraic (Hermite, 1873)

3 algebraic numbers Fact: Q is an algebraically closed subfield of C. In other words: 1. 0, 1 Q 2. if α, β Q, then α β, α + β, αβ Q 3. if furthermore β 0, then α/β Q 4. if f Q[x] is non-zero, then all complex roots of f lie in Q. Examples: Q all 5 complex zeros of 3x 5 + 7ix 3 2 lie in Q.

4 transcendental functions Let f : U C be complex differentiable on an open U C. Definition: f is transcendental if there are no a 0,..., a n C[x] with a n 0 such that for all z U: a n (z)f(z) n + + a 1 (z)f(z) + a 0 (z) = 0. If f is transcendental, then we do not expect f(z) Q for z Q. Example: if f(z) = z, then f is not transcendental, as f(z) 2 z = 0 for any non-zero α Q, we have exp(α) = e α Q (Lindemann, 1882)

5 algebraic values of transcendental functions But... sin( 7 11π) is a root of 1024x x x x x 2 11 Let θ(τ, z) = m Z qm2 Q m, where q = exp(2πiτ) and Q = exp(2πiz). Then θ(i, 0) θ(i, 1/3) = 3 2( 3 1). (e π ) 1/ = 1 12 (in a way)

6 one example explained Fact: Let e(z) = exp(2πiz). Then e(q) Q. Proof: Given z = a b Q, we have e(z)b = exp(2πia) = 1, so e(z) is a root of x b 1. Corollary: For z Q, we have cos(2πz) = 1 2 (e(z) + e( z)) Q, sin(2πz) = 1 2i (e(z) e( z)) Q. ζ 2 5 ζ ζ 5 = e(1/5) 1 ζ

7 the Galois group of a polynomial f The Galois group measures the symmetry of the roots of f. Historical application: It tells us whether the roots of f can be written in terms of radicals (i.e., n s) Definition: The Galois group Gal(f) of f is the set of permutations of the roots of f that preserve all polynomial relations over Q between them. Equivalently: The Galois group of f Q[x] is the set of field automorphisms of the splitting field L = Q({roots of f}) of f

8 the Galois group of a polynomial f Definition: The Galois group Gal(f) of f is the set of permutations of the roots of f that preserve all polynomial relations over Q between them. Example: let d Z non-square, and f = x 2 d choose a root d, the other is d L = {a + b d : a, b Q} Gal(f) has 2 elements: the identity map, and the map a + b d a b d, which switches the roots

9 abelian polynomials Definition: The Galois group Gal(f) of f is the set of permutations of the roots of f that preserve all polynomial relations over Q between them. Note: A random polynomial f of degree n usually does not have many relations between the roots, so usually Gal(f) = S n, the set of all n! permutations of the n complex roots. Definition: We call f and its roots abelian if Gal(f) is abelian (i.e., g, h Gal(f) : g h = h g). Let Q ab = {α Q : α is abelian} (is a field).

10 cyclotomic fields A larger example: ζ 5 = exp(2πi/5) is a root of f = x 4 + x 3 + x 2 + x + 1 = x5 1 x 1 and the other roots are ζ 2 5, ζ3 5, ζ4 5 any σ Gal(f) is completely determined by σ(ζ 5 ), because σ(ζ k 5 ) = σ(ζ 5) k σ(ζ 5 ) is a root of f, hence σ(ζ 5 ) = ζ a 5 with a {1, 2, 3, 4} ζ 5 = e(1/5) ζ ζ ζ 4 5

11 cyclotomic fields A larger example: ζ 5 = exp(2πi/5) is a root of f = x 4 + x 3 + x 2 + x + 1 = x5 1 x 1 and the other roots are ζ5 2, ζ3 5, ζ4 5 any σ Gal(f) is completely determined by σ(ζ 5 ), because σ(ζ5 k) = σ(ζ 5) k σ(ζ 5 ) is a root of f, hence σ(ζ 5 ) = ζ5 a with a {1, 2, 3, 4} composition of σ s is multiplication of a s mod 5. Bijection: (Z/5Z) Gal(f) (ζ a 5 )b = (ζ b 5 )a, so f is abelian. (a mod 5) (ζ 5 ζ a 5 ) More generally: Let ζ n = exp(2πi/n). Then ζ n is abelian with Galois group (Z/nZ) = {integers coprime to n taken modulo n }.

12 the Kronecker-Weber theorem Theorem: (Kronecker (1853), Weber (1886), Hilbert (1896)) If f Q[x] is abelian, then its roots lie in Q(ζ n ) for some n. Leopold Kronecker Heinrich Weber David Hilbert ( ) ( ) ( ) Example: For prime p 1 mod 4, we have p 1 2 More generally, n Z : n Q(ζ 4n ). k=1 ζk2 p = p.

13 the Kronecker-Weber theorem Theorem: (Kronecker (1853), Weber (1886), Hilbert (1896)) If f Q[x] is abelian, then its roots lie in Q(ζ n ) for some n. Example: For prime p 1 mod 4, we have p 1 2 More generally, n Z : n Q(ζ 4n ). In terms of complex functions: Q ab = Q(e(Q)) for e : z exp(2πiz) k=1 ζk2 p = p.

14 abelian extensions Now let K Q be any number field (e.g. Q( d)). Given f K[x], its Galois group over K is the set Gal(f/K) of permutations of the roots that preserve all polynomial relations over K between these roots. Equivalently, it is the set of automorphisms of the splitting field L of f over K that act trivially on K. Example: let f = x 3 2 = (x 3 2)(x ζ 3 3 2)(x ζ ) Gal(f/Q) = S 3 has order 6 Gal(f/Q(ζ 3 )) = {1, g, g 2 } with g : 3 2 ζ We call f and its roots abelian over K if Gal(f/K) is abelian.

15 the circle Recall: e(1/n) = exp(2πi/n) is abelian. Reason : addition formula for arc lengths on the circle: e(z 1 + z 2 ) = e(z 1 )e(z 2 ). repeated addition is multiplication: e(nz) = e(z) n. multiplication formulas are used in two ways: as division formula: 1 = e(n 1 n ) = e( 1 n )n, so e( 1 n ) is algebraic. as explicit Galois action: other roots e( a n ) = e( 1 n )a for a (Z/nZ)

16 the circle Can do the same in terms of sine, because of addition formulas: sin(z 1 + z 2 ) = sin (z 1 ) sin(z 2 ) + sin(z 1 ) sin (z 2 ) sin (z) = sin(z) sin (z) 2 = 1 sin(z) 2 Again: Repeatedly applying these rules gives multiplication formulas. They just look a bit more complicated. Example: ( ) sin(7z) = sin(z) 64 sin(z) sin(z) sin(z) 2 7 compare with e(7z) = e(z) 7

17 the lemniscate Let Q(i) = Q + Qi. Abel constructed elements of Q(i) ab by dividing arc lengths on the lemniscate. (x 2 + y 2 ) 2 = x 2 y 2 Niels Henrik Abel ( )

18 the sine The arc length on the unit circle from (1, 0) to (x, y) is (x, y 0, x 2 + y 2 = 1) z = y 0 1 du =: arcsin(y) 1 u 2 1 define sin(z) = y, and π = 2 arcsin(1) extend the sine to all of R. it is periodic with period 2πZ (i.e., k Z, z R : sin(z + 2πk) = sin(z))

19 the lemniscate sine For the arc length on the lemniscate (x 2 + y 2 ) 2 = x 2 y 2 from (0, 0) to (x, y), let r 2 = x 2 + y 2 and z = r 0 1 du =: arcsl(r). 1 u 4 define sl(z) = r, and let ω = 2arcsl(1) extend meromorphically to C 2ωZ-periodic notice sl(iz) = isl(z), so also 2iωZ-periodic

20 complex multiplication on the lemniscate Addition formula: (Euler (1751)) sl(z 1 + z 2 ) = sl(z 1)sl (z 2 ) + sl (z 1 )sl(z 2 ) 1 + sl(z 1 ) 2 sl(z 2 ) 2 Recall: sl(iz) = isl(z) Complex multiplication: For every a + bi Z[i] = Z + Zi, there is a multiplication formula for sl((a + bi)z) (Abel) Example: sl((2i 1)z) = sl(z) sl(z)4 + 2i 1 (2i 1)sl(z) 4 + 1

21 dividing lemniscate arc lengths Example: Dividing the lemniscate by 8, i.e., taking z = ω/4. r = sl(z) duplication formula gives r 4 + 2r 2 1 = 0, so r = 2 1. multiplication of z by ±1 and ±(2i 1) gives all roots: ± r and ± r r4 + 2i 1 (2i 1)r the polynomial x 4 + 2x 2 1 is abelian over Q[i], but not over Q In general: Galois groups of the form (Z[i]/(a + bi)). i

22 elliptic curves and elliptic functions The lemniscate sine is periodic with respect to a lattice in C. It links that lattice to the lemniscate. Elliptic functions link any lattice Λ C to an elliptic curve y 2 = x 3 + ax + b. Scale and rotate so that Λ = Z + τz with Im(τ) > 0. Weierstrass s -function (τ, z) = 1 z 2 + is Λ-periodic in z, m,n Z (m,n) (0,0) ( ) 1 (z + m + nτ) 2 1 (m + nτ) 2 has transformation and addition laws for τ and z

23 Kronecker s Jugendtraum Let K be imaginary quadratic (i.e., K = Q( d), d < 0). If τ K, then there exist complex multiplication formulas for z (τ, z) by elements of K. Get abelian numbers over K: (let 2w = max{n : ζ n K}) (τ, z 1 ) w (τ, z 2 ) w, τ, z 1, z 2 K. Kronecker in a letter to Dedekind (1880):... meinem liebsten Jugendtraum, nämlich... den Nachweis, dass die Abel schen Gleichungen mit Quadratwurzeln rationaler Zahlen durch die Transformationsgleichungen elliptischer Funktionen mit singulären Moduln gerade so erschöpft werden, wie die ganzzahligen Abel schen Gleichungen durch die Kreistheilungsgleichungen.

24 modular functions, the j-invariant Recall Λ = Z + τz and is a function in τ and z. Modular functions depend only on τ. The j-invariant is such a function in τ, and in fact is a bijection {lattices in C up to scaling and rotation} C We have j(τ) K ab for τ K imaginary quadratic. Hilbert seemed to think that Kronecker conjectured K ab = K(j(K), e(q)).

25 solution to the Jugendtraum Many tried to prove what Hilbert suggested: K ab = K(j(K), e(q)), until Fueter in 1914 proved α = i K(j(K), e(q)) for K = Q(i). α K ab because roots of x 4 (1 + 2i) are i k α for k Z/4Z. After that, Kronecker s Jugendtraum could be fulfilled: given τ K and z 0 Q with ν = (τ, z 0 ), we have K ab = K((ν 1 (τ, Q)) w ).

26 Hilbert s 12th problem At the International Congress of Mathematicians in Paris in 1900, David Hilbert presented a list of 23 problems. Number 12 is to extend the Kronecker-Weber theorem Q ab = Q(exp(2πiQ)) to arbitrary base fields K instead of Q. In other words, given any number field K, 1. to classify the abelian Galois groups over K, and 2. to find complex functions analogous to e : z exp(2πiz) and j to generate K ab. Hilbert: I regard this problem as one of the most profound and far-reaching in the theory of numbers and of functions.

27 class field theory Part 1 asked for a classification of the abelian extensions of any number field K. The answer to that is class field theory (Takagi, Artin, many others ( )) Part 2 asks for a complex analytic construction. solved for K = Q (Kronecker-Weber theorem) solved for imaginary quadratic fields (Kronecker s Jugendtraum) partially solved for CM-fields (complex multiplication of abelian varieties, Shimura and Taniyama, 1950 s)

28 abelian varieties and theta functions Abelian varieties generalize elliptic curves to any dimension. Theta functions generalize, and make them computationally more efficient. Let τ Mat g (C) be symmetric with positive definite imaginary part, and let c 1, c 2 R g and z C g. Then θ[c 1, c 2 ](τ, z) = n Z g exp(πi(n+c 1 ) t τ(n+c 1 )+2πi(n+c 1 ) t (z+c 2 )). Example: (g = 1, c 1 = c 2 = 0) θ(i, 0) θ(i, 1/3) = 3 2( 3 1) is abelian over Q(i).

29 = 1/12 The Riemann zeta function is given by ζ(s) = n=1 1 n s = p prime (1 p s ) 1 if Re(s) > 1. It is analytically continued to C \ {1}, and has e.g. ζ( 1) = Properties of ζ give information about primes. L-functions are variations on the definition ζ.

30 Stark s conjectures Stark s conjectures (1970 s) predict special algebraic values of certain L-functions. Some cases are proven using exp(2πiq) or complex multiplication of elliptic curves. All other cases are still unproven are arguably a solution to Hilbert s 12th problem hint at unkown structure Given a field F = Q( d), but this time with d > 0, the mathematics software PARI/GP successfully uses Stark s conjectures to construct (proven) real abelian extensions.

31 exp(π 163) 744 (640320) 3 Given K = Q( d) with d < 0 square-free, let ω = d if d 3 mod 4 and ω = ( d + 1)/2 otherwise. Kronecker called K(j(ω)) the species of K (now called the Hilbert class field). Let q = exp(2πiτ). Then j(τ) = q q q 2 +. Finitely many d with j(ω) Q, the largest in absolute value is 163. Get q = e π 163, so q is very small, so e π = q j(ω) Z. Not just any integer: it is a cube!

32 the cube root of j to compute the Hilbert class field, small generators help for K = Q( 5), we have j(ω) = ( ) 3 j(ω) is a root of x x j(ω) 1/3 is a root of x 2 100x 880 (smaller!) using explicit Galois actions, can prove in advance j(ω) 1/3 K(j(ω))

33 class invariants for genus 2 (own work) By making Galois action for g = 2 explicit, can reduce x x x x x x x x x x 10 to 4096y ( 5 + ) 14336α α α y 4 + ( ) 37504α α α y 3 + ( ) 43408α α α y 2 + ( ) 20024α α α y+ 87α α α The roots of both polynomials generate the same degree-5 extension of K = Q(α) with α α = 0.

34 References Keith Conrad, History of Class Field Theory Norbert Schappacher, On the History of Hilbert s Twelfth Problem, A Comedy of Errors, Matériaux pour l histoire des mathématiques au XXe siècle (Nice, 1996), Franz Lemmermeyer, Reciprocity Laws, from Euler to Eisenstein, Springer 2000